Druyd:Knowledge

Oscillators of a Spring

Storyboard

In the case of the spring the force is proportional to the elongation of the spring so that the equations of motion are linear and the frequency of the oscillation is independent of the amplitude. This is the key to generate an oscillation that does not depend on the fact that the friction decreases over time. This is why old clocks used (circular) springs to generate stable oscillations to measure elapsed time.

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ID:(1425, 0)

Mechanisms

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Knowledge

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Concept

Mechanisms

ID:(15848, 0)

Oscillations with a spring

Description

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One of the systems it depicts is that of a spring. This is associated with the elastic deformation of the material from which the spring is made. By "elastic," we mean a deformation that, upon removing the applied stress, allows the system to fully regain its original shape. It's understood that it doesn't undergo plastic deformation.

Since the energy of the spring is given by

$E=\displaystyle\frac{1}{2}m_i v^2+\displaystyle\frac{1}{2}k x^2$

the period will be equal to

$T=2\pi\sqrt{\displaystyle\frac{m_i}{k}}$

and thus, the angular frequency is

$\omega_0 ^2=\displaystyle\frac{ k }{ m_i }$

ID:(15563, 0)

Model

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Parameters

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$x$

Elongation of the Spring

$\omega$

omega

Frecuencia angular del resorte

rad/s

$k$

Hooke Constant

N/m

$m_i$

m_i

Inertial Mass

$x_0$

x_0

Initial amplitude of the oscillation

$v$

Oscillator speed

m/s

$\pi$

rad

$V$

Potential Energy

$E$

Total Energy

$K$

Total Kinetic Energy

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$\nu$

Frequency

$p$

Moment

kg m/s

$T$

Period

$t$

Time

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

Equation

$E = K + V$

E = K + V

$K =\displaystyle\frac{ p ^2}{2 m_i }$

K = p ^2/(2 * m_i )

$\nu =\displaystyle\frac{1}{ T }$

nu =1/ T

$\omega = 2 \pi \nu$

omega = 2* pi * nu

$\omega = \displaystyle\frac{2 \pi }{ T }$

omega = 2* pi / T

$\omega_0 ^2=\displaystyle\frac{ k }{ m_i }$

omega_0 ^2 = k / m_i

$p = m_i v$

p = m_i * v

$T =2 \pi \sqrt{\displaystyle\frac{ m_i }{ k }}$

T =2* pi *sqrt( m_i / k )

$v = - x_0 \omega_0 \sin \omega_0 t$

v = - x_0 * omega_0 *sin( omega_0 * t )

$V =\displaystyle\frac{1}{2} k x ^2$

V = k * x ^2/2

$E =\displaystyle\frac{1}{2} k x_0 ^2$

V = k * x ^2/2

$x = x_0 \cos \omega t$

x = x_0 *cos( omega_0 * t )

ID:(15851, 0)

Total Energy

Equation

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The total energy corresponds to the sum of the total kinetic energy and the potential energy:

$E = K + V$

$V$

Potential Energy

$J$

4981

$E$

Total Energy

$J$

5290

$K$

Total Kinetic Energy

$J$

5314

ID:(3687, 0)

Kinetic energy as a function of moment

Equation

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The kinetic energy of a mass $m$

$K_t =\displaystyle\frac{1}{2} m_i v ^2$

can be expressed in terms of momentum as

$K =\displaystyle\frac{ p ^2}{2 m_i }$

$m_i$

Inertial Mass

$kg$

6290

$p$

Moment

$kg m/s$

8974

$K$

Total Kinetic Energy

$J$

5314

Since kinetic energy is equal to

$K_t =\displaystyle\frac{1}{2} m_i v ^2$

and momentum is

$p = m_i v$

we can express it as

$K_t=\displaystyle\frac{1}{2} m_i v^2=\displaystyle\frac{1}{2} m_i \left(\displaystyle\frac{p}{m_i}\right)^2=\displaystyle\frac{p^2}{2m_i}$

$K =\displaystyle\frac{ p ^2}{2 m_i }$

ID:(4425, 0)

Elastic potential energy (1)

Equation

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En el caso elástico (resorte) la fuerza es

la energía

$dW = \vec{F} \cdot d\vec{s}$

se puede mostrar que en este caso es

$V =\displaystyle\frac{1}{2} k x ^2$

$x$

Elongation of the Spring

$m$

5313

$k$

Hooke Constant

$N/m$

5311

$V$

Potential Energy

$J$

4981

En el caso elástico (resorte) la fuerza es

con k la constante del resorte y x la elongación/compresión del resorte. La variación de la energía potencial es

$dW = \vec{F} \cdot d\vec{s}$

\\n\\nLa diferencia\\n\\n

$\Delta x = x_2 - x_1$

\\n\\ncorresponde al camino recorrido por lo que\\n\\n

$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$

y con ello la energía potencial elástica es

$V =\displaystyle\frac{1}{2} k x ^2$

ID:(3246, 1)

Elastic potential energy (2)

Equation

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En el caso elástico (resorte) la fuerza es

la energía

$dW = \vec{F} \cdot d\vec{s}$

se puede mostrar que en este caso es

$E =\displaystyle\frac{1}{2} k x_0 ^2$

$V =\displaystyle\frac{1}{2} k x ^2$

$x$

$x_0$

Initial amplitude of the oscillation

$m$

9961

$k$

Hooke Constant

$N/m$

5311

$V$

$E$

Total Energy

$J$

5290

En el caso elástico (resorte) la fuerza es

con k la constante del resorte y x la elongación/compresión del resorte. La variación de la energía potencial es

$dW = \vec{F} \cdot d\vec{s}$

\\n\\nLa diferencia\\n\\n

$\Delta x = x_2 - x_1$

\\n\\ncorresponde al camino recorrido por lo que\\n\\n

$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$

y con ello la energía potencial elástica es

$V =\displaystyle\frac{1}{2} k x ^2$

ID:(3246, 2)

Oscillations with a spring

Equation

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The product of the hooke Constant ( $k$ ) and the inertial Mass ( $m_i$ ) is called the frecuencia angular del resorte ( $\omega$ ) and is defined as:

$\omega_0 ^2=\displaystyle\frac{ k }{ m_i }$

$\omega_0$

Frecuencia angular del resorte

$rad/s$

9798

$k$

Hooke Constant

$N/m$

5311

$m_i$

Inertial Mass

$kg$

6290

ID:(1242, 0)

Moment

Equation

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The moment ( $p$ ) is calculated from the inertial Mass ( $m_i$ ) and the speed ( $v$ ) using

$p = m_i v$

$m_i$

Inertial Mass

$kg$

6290

$p$

Moment

$kg m/s$

8974

$v$

Oscillator speed

$m/s$

9965

ID:(10283, 0)

Periodo de la Oscilación

Equation

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Como la oscilación cumple las leyes físicas se puede hacer uso del hecho que el area debajo de la curva velocidad vs tiempo el camino recorrido para determinar el perido. Como la velocidad es\\n\\n

$\displaystyle\int_0^{T/2}v(t)dt=\sqrt{\displaystyle\frac{2E}{m}}\displaystyle\int_0^{T/2}\cos \displaystyle\frac{2\pi t}{T}dt=\sqrt{\displaystyle\frac{2E}{m}}\displaystyle\frac{T}{\pi}$

\\n\\ny el camino entre un mínimo a un máximo de una elongación, lo que ocurre entre el tiempo 0 y T/2 es igual a\\n\\n

$x_{max}-x_{min}=2\sqrt{\displaystyle\frac{2E}{k}}$

se tiene que

$T =2 \pi \sqrt{\displaystyle\frac{ m_i }{ k }}$

$k$

Hooke Constant

$N/m$

5311

$m_i$

Inertial Mass

$kg$

6290

$T$

Period

$s$

5078

$\pi$

3.1415927

$rad$

5057

ID:(7106, 0)

Frequency

Equation

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The frequency ( $\nu$ ) corresponds to the number of times an oscillation occurs within one second. The period ( $T$ ) represents the time it takes for one oscillation to occur. Therefore, the number of oscillations per second is:

$\nu =\displaystyle\frac{1}{ T }$

$\nu$

Frequency

$Hz$

5077

$T$

Period

$s$

5078

Frequency is indicated in Hertz (Hz).

ID:(4427, 0)

Angular frequency

Equation

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The angular frequency ( $\omega$ ) is with the period ( $T$ ) equal to

$\omega = \displaystyle\frac{2 \pi }{ T }$

$\omega$

Frecuencia angular del resorte

$rad/s$

9798

$T$

Period

$s$

5078

$\pi$

3.1415927

$rad$

5057

ID:(12335, 0)

Relación frecuencia angular - frecuencia

Equation

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Como la frecuencia angular es con angular frequency $rad/s$ , period $s$ and pi $rad$ igual a

$\omega = \displaystyle\frac{2 \pi }{ T }$

y la frecuencia con frequency $Hz$ and period $s$ igual a

$\nu =\displaystyle\frac{1}{ T }$

se tiene que con frequency $Hz$ and period $s$ igual a

$\omega = 2 \pi \nu$

$\omega$

Frecuencia angular del resorte

$rad/s$

9798

$\nu$

Frequency

$Hz$

5077

$\pi$

3.1415927

$rad$

5057

ID:(12338, 0)

Oscillation amplitude

Equation

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With the description of the oscillation using

$z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t$

the real part corresponds to the temporal evolution of the amplitude

$x = x_0 \cos \omega t$

$x = x_0 \cos \omega_0 t$

$x_0$

Initial amplitude of the oscillation

$m$

9961

$x$

Elongation of the Spring

$m$

5313

$\omega_0$

$\omega$

Frecuencia angular del resorte

$rad/s$

9798

$t$

Time

$s$

5264

ID:(14074, 0)

Swing speed

Equation

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When we extract the real part of the derivative of the complex number representing the oscillation

$\dot{z} = i \omega_0 z$

whose real part corresponds to the velocity

$v = - x_0 \omega \sin \omega t$

$v = - x_0 \omega_0 \sin \omega_0 t$

$\omega_0$

Frecuencia angular del resorte

$rad/s$

9798

$x_0$

Initial amplitude of the oscillation

$m$

9961

$v$

Oscillator speed

$m/s$

9965

$t$

Time

$s$

5264

Using the complex number

$z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t$

introduced in

$\dot{z} = i \omega_0 z$

we obtain

$\dot{z} = i\omega_0 z = i \omega_0 x_0 \cos \omega_0 t - \omega_0 x_0 \sin \omega_0 t$

thus, the velocity is obtained as the real part

$v = - x_0 \omega_0 \sin \omega_0 t$

ID:(14076, 0)